 Gravitational focusing

Gravitational focusing enlarges the collision cross-section of gravitating bodies because attraction bends incoming trajectories toward one another. Let two spherical bodies have total mass $M$, sum of radii $R$, and relative speed $v_{\infty }$ at large separation. In the Newtonian two-body problem, an encounter that just grazes at separation $R$ satisfiesHere $b_{\max }$ is the limiting impact parameter. Conservation of energy sets the speed at closest approach, and conservation of angular momentum relates that speed to the initial impact parameter.

The extra term is the squared ratio of the mutual escape velocity at contact to $v_{\infty }$. Rapid encounters approach the geometric area $\pi R^2$, while slower encounters can have a substantially larger collision cross-section. This estimate assumes isolated two-body trajectories and counts physical contact rather than permanent gravitational capture.